Should be able to get LAN to check your bags through, just make sure that the agent in CUZ checks them through. You'll need to remind them it's a Oneworld to Oneworld connection if they have a problem. You might be a tight if the LAN flight gets very much of a delay since you'll need to clear immigration and sometimes it can be a zoo but I don't remember to much of a delay at that time in the morning. Just make sure you've got a BP, it might be best to try to print out a hard copy for immigration since I haven't used a mobile BP in Lima. It's definitely closer than I'd do especially in LIM. Good Luck!